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The sum complex $X_{A,k}$ associated to a subset $A \\subset G$ and $k < n$, is the $k$-dimensional simplicial complex obtained by taking the full $(k-1)$-skeleton of $\\Delta_{n-1}$ together with all $(k+1)$-subsets $\\sigma \\subset G$ that satisfy $\\sum_{x \\in \\sigma} x \\in A$. Let $C^{k-1}(X_{A,k})$ denote the space of complex valued $(k-1)$-cochains of $X_{A,k}$. Let $L_{k-1}:C^{k-1}(X_{A,k}) \\rightarrow C^{k-1}(X_{A,k})$ denote the reduced $(k-1)$-th Laplacian of $X_{A,k}$,"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1801.06466","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-19T15:39:59Z","cross_cats_sorted":[],"title_canon_sha256":"66ad2188ce3abf4ac27af685042a9a7020f74b90663f7671418b02786b32a657","abstract_canon_sha256":"303228153f1bc4aa229a6de5d988d2e8e43db99ee8ad26ae12c40b737b69e547"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:29.972298Z","signature_b64":"DdSW2AZx0PLjynu8cIOuQ9pR7fnQ0gt0pf4Ken+DV3DCUs6JznLfnVslS1uqn/sgnn2Rstax/b36JgS94TnqCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11e28d4773cf1d8056ce41164942d559071b673d1efa83b5e15617d2e664a9b4","last_reissued_at":"2026-05-18T00:25:29.971643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:29.971643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral expansion of random sum complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Orr Beit-Aharon, Roy Meshulam","submitted_at":"2018-01-19T15:39:59Z","abstract_excerpt":"Let $G$ be a finite abelian group of order $n$ and let $\\Delta_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex $X_{A,k}$ associated to a subset $A \\subset G$ and $k < n$, is the $k$-dimensional simplicial complex obtained by taking the full $(k-1)$-skeleton of $\\Delta_{n-1}$ together with all $(k+1)$-subsets $\\sigma \\subset G$ that satisfy $\\sum_{x \\in \\sigma} x \\in A$. Let $C^{k-1}(X_{A,k})$ denote the space of complex valued $(k-1)$-cochains of $X_{A,k}$. Let $L_{k-1}:C^{k-1}(X_{A,k}) \\rightarrow C^{k-1}(X_{A,k})$ denote the reduced $(k-1)$-th Laplacian of $X_{A,k}$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06466","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.06466","created_at":"2026-05-18T00:25:29.971739+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.06466v1","created_at":"2026-05-18T00:25:29.971739+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.06466","created_at":"2026-05-18T00:25:29.971739+00:00"},{"alias_kind":"pith_short_12","alias_value":"CHRI2R3TZ4OY","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"CHRI2R3TZ4OYAVWO","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"CHRI2R3T","created_at":"2026-05-18T12:32:16.446611+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE","json":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE.json","graph_json":"https://pith.science/api/pith-number/CHRI2R3TZ4OYAVWOIELESQWVLE/graph.json","events_json":"https://pith.science/api/pith-number/CHRI2R3TZ4OYAVWOIELESQWVLE/events.json","paper":"https://pith.science/paper/CHRI2R3T"},"agent_actions":{"view_html":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE","download_json":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE.json","view_paper":"https://pith.science/paper/CHRI2R3T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.06466&json=true","fetch_graph":"https://pith.science/api/pith-number/CHRI2R3TZ4OYAVWOIELESQWVLE/graph.json","fetch_events":"https://pith.science/api/pith-number/CHRI2R3TZ4OYAVWOIELESQWVLE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE/action/storage_attestation","attest_author":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE/action/author_attestation","sign_citation":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE/action/citation_signature","submit_replication":"https://pith.science/pith/CHRI2R3TZ4OYAVWOIELESQWVLE/action/replication_record"}},"created_at":"2026-05-18T00:25:29.971739+00:00","updated_at":"2026-05-18T00:25:29.971739+00:00"}